数据分析83Word格式.docx

数据分析83Word格式.docx

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1.00

394=n−k−1=398−3−1

398

395

2

3

Total

1.00/1.00

QuestionExplanationThisquestionreferstothefollowinglearningobjective（s）:

Notethatthep-valuesassociatedwitheachpredictorareconditionalonothervariablesbeingincludedinthemodel,sotheycanbeusedtoassessifagivenpredictorissignificant,giventhatallothersareinthemodel.Thesep-valuesarecalculatedbasedonatdistributionwithn−k−1degreesoffreedom.

Question2

Arandomsampleof200womenwhowereatleast21yearsold,ofPimaIndianheritageandlivingnearPhoenix,Arizona,weretestedfordiabetesaccordingtoWorldHealthOrganizationcriteria.Themodelbelowisusedforpredictingtheirplasmaglucoseconcentrationbasedontheirdiastolicbloodpressure（bp,inmmHg）,age（age,inyears）,andwhetherornottheyarediabetic（type,YesandNo）.Whatisthepredictedbloodglucoselevelofa30yearoldwomanwhohasadiastolicbloodpressureof72mmHgandisnotdiabetic?

38.1

140.67

117.46

114.1

Definethemultiplelinearregressionmodelas

y^=β0+β1x1+β2x2+⋅⋅⋅+βkxk

wheretherearekpredictors（explanatoryvariables）.

Question3

Bodyfatpercentagecanbecomplicatedtoestimate,whilevariablessuchasAge,Height,Weight,andmeasurementsofvariousbodypartsareeasytomeasure.Basedondata1onbodyfatpercentageandothervariousmeasurements,alinearregressionmodelwasdevelopedtopredictbodyfatpercentage,basedoneasytoobtainmeasurements.Themodeloutputisshownbelow.

Basedonthisoutput,whatisthecorrectinterpretationofthecoefficientforwrist?

1Penrose,K.,Nelson,A.,andFisher,A.（1985）,GeneralizedBodyCompositionPredictionEquationforMenUsingSimpleMeasurementTechniques,MedicineandScienceinSportsandExercise,7

（2）,189.

Forevery1inchincreaseinwristcircumference,we’dexpectadecreaseinbodyfatpercentageofabout1.14%plusorminus1.96∗0.47%,onaverage.

Forevery1inchdecreaseinwristcircumference,we’dexpectadecreaseinbodyfatpercentageof1.14%,onaverage.

Forevery1inchdecreaseinwristcircumference,we’dexpectadecreaseinbodyfatpercentageofabout（13.57+1.14）%,onaverage.

Forevery1inchincreaseinwristcircumference,wewouldexpectbodyfatpercentagetobelowerby1.14%,onaverage.

  -Interprettheestimatefortheintercept（b0）astheexpectedvalueofywhenallpredictorsareequalto0,onaverage.

  -Interprettheestimateforaslope（sayb1）as“Allelseheldconstant,foreachunitincreaseinx1,wewouldexpectytobehigher/loweronaveragebyb1.”

Question4

Wemodeledthepricesof93cars（in$1,000s）usingitscityMPG（milespergallon）anditsmanufacturingsite（foreignordomestic）.Theregressionoutputisprovidedbelow.Notethatdomesticisthereferencelevelformanufacturingsite.Dataareoutdatedsothepricesmayseemlow.

Whichofthefollowingisfalse?

CityMPGisasignificantpredictorofcarprice,giveninformationonthemanufacturingsiteofthecar.

Manufacturingsiteisasignificantpredictorofcarprice,giveninformationonthecityMPGofthecar.

Inorrect

0.00

Thep-valueforsiteislow,henceit’sasignificantpredictorinthismodel.

The95%confidenceintervalfortheslopeofcityMPGcanbecalculatedas−1.14±

（−8.03∗0.14）.

Ifweaddanothervariabletothemodel,sayhighwayMPG,thep-valuesassociatedwithcityMPGandmanufacturingsitemaychange.

0.00/1.00

  -ThesignificanceofthemodelasawholeisassessedusinganF-test.

   -H0:

β1=β2=⋅⋅⋅=βk

   HA:

Atleastoneβi≠0.

   -df=n−k−1degreesoffreedom.

   -Usuallyreportedatthebottomoftheregressionoutput.

-Notethatthep-valuesassociatedwitheachpredictorareconditionalonothervariablesbeingincludedinthemodel,sotheycanbeusedtoassessifagivenpredictorissignificant,giventhatallothersareinthemodel.

  -Thesep-valuesarecalculatedbasedonatdistributionwithn−k−1degreesoffreedom.

  -Thesamedegreesoffreedomcanbeusedtoconstructaconfidenceintervalfortheslopeparameterofeachpredictor:

      bi±

t⋆n−k−1SEbi

Question5

R2willneverdecreasewhenapredictorisaddedtoalinearmodel.

False

True

NotethatR2willincreasewitheachexplanatoryvariableaddedtothemodel,regardlessofwhetherornottheaddedvariableisameaningfulpredictoroftheresponsevariable.ThereforeweuseadjustedR2,whichappliesapenaltyforthenumberofpredictorsincludedinthemodel,tobetterassessthestrengthofamultiplelinearregressionmodel:

      R2adj=1−SSE/（n−k−1）SST/（n−1）

wherenisthenumberofcasesandkisthenumberofpredictors.

  -NotethatR2adjwillonlyincreaseiftheaddedvariablehasameaningfulcontributiontotheamountofexplainedvariabilityiny,i.e.ifthegainsfromaddingthevariableexceedsthepenalty.

Question6

Considerthefollowingoutputfromamultiplelinearregressionmodelwith10predictors.

Ifyouweredoingbackwardsselectiononthismodelusingp-valueasthecriterion,whichofthefollowingwouldbeanacceptablenextstep?

Removethevariable“dollar”becauseithasthehighestp-value.

Removeoneofthevariables“resubj”or“attach”becausetheybothhavethelowestp-values.

Inbackwardsselectionusingp-valueasthecriterion,wewanttokeepvariableswithlowp-valuesandremovethevariablewiththehighestp-value.

Removeanyoneofthevariableswithahighp-valuewhichaslongasremovingthevariabledoesnotcausetheadjustedR2todecreaseinthere-fittedmodel.

Removethevariables“password”and“dollar”becausetheirhighp-valuesindicatecollinearitywithothervariables.

Thegeneralideabehindbackward-selectionistostartwiththefullmodelandeliminateonevariableatatimeuntiltheidealmodelisreached.

  -p-valuemethod:

   （i）Startwiththefullmodel.

   （ii）Dropthevariablewiththehighestp-valueandrefitthemodel.

   （iii）Repeatuntilallremainingvariablesaresignificant.

  -adjustedR2method:

   （ii）Refitallpossiblemodelsomittingonevariableatatime,andchoosethemodelwiththehighestadjustedR2.

   （iii）RepeatuntilmaximumpossibleadjustedR2isreached.

Question7

AspartoftheSecondInternationalMathematicsStudyon8thgradersfromrandomlysam-pledclassroomsintheUSwhocompletedmathematicsachievementtestsatthebeginningandattheendoftheacademicyear.Studentsalsoansweredquestionersregardingtheirattitudestowardmathematics.Thelinearmodeloutputpredictsthegainscoreinthistest（post-test-pretestscore）usingthefollowingexplanatoryvariables:

  •pretest:

scoreontheexamtakenatthebeginningofthesemester

  •gender:

maleorfemale

  •more_ed:

expectednumberofyearsforcontinuededucation（upto2years,2to5years,5to6years,8ormoreyears）

  •useful:

Mathisusefulineverydaylife（stronglydisagree,disagree,undecided,agree,stronglyagree）

  •ethnic:

ethnicityofstudent（AfricanAmerican,Anglo,Other）

Thefollowingistheresidualsplotforthismodel.Whichofthefollowingconditionscanthisplotbeusedtocheck?

Independentresiduals

NearlynormalresidualsFeedback:

Constantvariabilityofresiduals

Non-collinearexplanatoryvariables.

Requirespairwisescatterplotforeachcombinationofexplanatoryvariables.

Listtheconditionsformultiplelinearregressionas

（1）linearrelationshipbetweeneach（numerical）explanatoryvariableandtheresponse-checkedusingscatterplotsofyvs.eachx,andresidualsplotsofresidualsvs.eachx

（2）nearlynormalresidualswithmean0-checkedusinganormalprobabilityplotandhistogramofresiduals

（3）constantvariabilityofresiduals-checkedusingresidualsplotsofresidualsvs.y^,andresidualsvs.eachx

（4）independenceofresiduals（andhenceobservations）-checkedusingascatterplotofresidualsvs.orderofdatacollection（willrevealnon-independenceifda